ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  moimv Unicode version

Theorem moimv 2043
Description: Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
Assertion
Ref Expression
moimv  |-  ( E* x ( ph  ->  ps )  ->  ( ph  ->  E* x ps )
)
Distinct variable group:    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem moimv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ax-1 6 . . . . . . 7  |-  ( ps 
->  ( ph  ->  ps ) )
21a1i 9 . . . . . 6  |-  ( ph  ->  ( ps  ->  ( ph  ->  ps ) ) )
32sbimi 1722 . . . . . . 7  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] ( ps  ->  (
ph  ->  ps ) ) )
4 nfv 1493 . . . . . . . 8  |-  F/ x ph
54sbf 1735 . . . . . . 7  |-  ( [ y  /  x ] ph 
<-> 
ph )
6 sbim 1904 . . . . . . 7  |-  ( [ y  /  x ]
( ps  ->  ( ph  ->  ps ) )  <-> 
( [ y  /  x ] ps  ->  [ y  /  x ] (
ph  ->  ps ) ) )
73, 5, 63imtr3i 199 . . . . . 6  |-  ( ph  ->  ( [ y  /  x ] ps  ->  [ y  /  x ] (
ph  ->  ps ) ) )
82, 7anim12d 333 . . . . 5  |-  ( ph  ->  ( ( ps  /\  [ y  /  x ] ps )  ->  ( (
ph  ->  ps )  /\  [ y  /  x ]
( ph  ->  ps )
) ) )
98imim1d 75 . . . 4  |-  ( ph  ->  ( ( ( (
ph  ->  ps )  /\  [ y  /  x ]
( ph  ->  ps )
)  ->  x  =  y )  ->  (
( ps  /\  [
y  /  x ] ps )  ->  x  =  y ) ) )
1092alimdv 1837 . . 3  |-  ( ph  ->  ( A. x A. y ( ( (
ph  ->  ps )  /\  [ y  /  x ]
( ph  ->  ps )
)  ->  x  =  y )  ->  A. x A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
) )
11 ax-17 1491 . . . 4  |-  ( (
ph  ->  ps )  ->  A. y ( ph  ->  ps ) )
1211mo3h 2030 . . 3  |-  ( E* x ( ph  ->  ps )  <->  A. x A. y
( ( ( ph  ->  ps )  /\  [
y  /  x ]
( ph  ->  ps )
)  ->  x  =  y ) )
13 ax-17 1491 . . . 4  |-  ( ps 
->  A. y ps )
1413mo3h 2030 . . 3  |-  ( E* x ps  <->  A. x A. y ( ( ps 
/\  [ y  /  x ] ps )  ->  x  =  y )
)
1510, 12, 143imtr4g 204 . 2  |-  ( ph  ->  ( E* x (
ph  ->  ps )  ->  E* x ps ) )
1615com12 30 1  |-  ( E* x ( ph  ->  ps )  ->  ( ph  ->  E* x ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1314   [wsb 1720   E*wmo 1978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator